Transmission channels such as, e.g., telephone connections, microwave systems, and radio links are subject to a variety of frequency dependent intrinsic impairments or extraneous disturbances. For example, in a voiceband telephone channel, impairments such as amplitude attenuation and phase distortion are relatively low at frequencies in the range of approximately 0.4-2.8 kHz but exhibit rapid increase at frequencies below and above this range. Consequently, in order to avoid undue distortion of signals during passage through a telephone channel, transmission of voice or data preferably is limited to frequencies in a suitably restricted range.
While restriction of frequency range is virtually mandated by inherent physical limitations of the typical telephone channel, such restriction may be used to economic advantage in systems which are physically less severely limited. For example, in a microwave or optical communications system, so-called frequency division multiplexing permits the simultaneous operation of several communications channels over one and the same physical facility when non-overlapping portions of the available spectrum are assigned to different communications channels. It is evident that minimization of frequency range assigned to individual channels may lead to an increase in the number of multiplexed channels.
In the case of data communications, there is an additional and paramount concern with transmission rate as limited by bandwidth. From the theoretical work of H. Nyquist it is well known that, in order to transmit numbers at a rate .rho. per second by pulse amplitude modulation, bandwidth of at least .rho./2 Hertz is required. Moreover, use of such minimal bandwidth, conveniently designated Nyquist bandwidth, depends on a number of idealizing assumptions regarding pulse shape and channel properties whose physical implementation is not practical. Pulse shape, in particular, is ideally required to have a perfectly rectangular spectrum of constant nonzero amplutide for frequencies within the frequency band and constant zero amplitude for frequencies outside the band. Such ideal spectrum corresponds to a pulse of the form (sin x)/x which may be approximated but not exactly realized in practice.
Physically realizable pulses which are suitable for pulse amplitude modulation are shown, e.g., in the book by R. W. Lucky et al., "Principles of Data Communications", McGraw-Hill, 1968 on page 51, in the book by William R. Bennett et al., "Data Transmission.revreaction., McGraw-Hill, 1965 on page 56, and in U.S. Pat. No. 2,719,189 (issued on Sept. 27, 1955 to W. R. Bennett et al., "Prevention of Interpulse Interference in Pulse Multiplex Transmission"). Such pulses are characterized by a frequency spectrum which has a frequency interval in which amplitude is a nonzero constant and a so-called roll-off interval of frequencies at which amplitude decreases to zero smoothly and symmetrically with respect to the halfway point. Resulting pulses are known as Nyquist pulses and, at the price of additional bandwidth, are actually superior for purposes of data transmission based on pulse amplitude modulation. An often used Nyquist pulse is the so-called raised cosine pulse shown in the references cited above.
In a conventional pulse amplitude modulated data transmission system, data symbols .alpha..sub.j having 2.sup.L values or levels are represented by pulses whose amplitude at time jT is directly proportional to .alpha..sub.j, T being a fixed time interval between pulses transmitted. For example, if L=1, then, typically, the .alpha..sub.j 's are +1 or -1. Since the resulting signal is a linear combination of time translates of a single pulse, the bandwidth of the signal is equal to the bandwidth of the pulse which, ideally, can be as narrow as 1/(2T). However, due to roll-off as described above, an "excess bandwidth" of at least 10-20 percent is required in practice. In the case of a telephone channel, for example, transmission can be achieved at a rate of 9.6.times.10.sup.3 bits/second using 4-level pulse amplitude modulated signals and corresponding to an ideal bandwidth of 2.4 kHz. However, allowing for 12 percent excess bandwidth, the transmitted signal occupies a frequency range which extends from approximately 0.36 to approximately 3.05 kHz.
While it is possible to transmit data items .alpha..sub.j one at a time, actual transmission systems often use some form of data encoding based on groups of a fixed number .nu. of data items. For example, a simple encoding scheme used in error detection consists in forming the sum (modulo 2) of a block of .nu. binary data items and appending it as an (n+1)-st item to the block. Thus, for each .nu. data items, N=.nu.+1 signals are transmitted and, at the expense of a slight reduction in data rate, information is supplied to the receiver which allows the detection of a transmission error.
Encoding may be used for purposes other than error detection such as, for example, to convert a 2-level signal into a 3-level signal in duobinary systems disclosed on pages 83-88 of the book by Lucky et al. cited above. Also, according to U.S. Pat. No. 3,388,330 (issued June 11, 1968 to E. R. Kretzmer) and as described on pages 88-92 of the book by Lucky et al., encoding may be used to achieve desirable frequency characteristics in so-called partial response systems.